A calculational approach to program inversion

Many problems in computation can be specified in terms of computing the inverseof an easily constructed function. However, studies on how to derive an algorithm froma problem specification involving inverse functions are relatively rare. The aim of thisthesis is to demonstrate, in an example-driven style, a number of techniques to do the job.The techniques are based on the framework of relational, algebraic program derivation.Simple program inversion can be performed by just taking the converse of the program,sometimes known as to “run a program backwards”. The approach, however, does notmatch the pattern of some more advanced algorithms. Previous results, due to Bird andde Moor, gave conditions under which the inverse of a total function can be written asa fold. In this thesis, a generalised theorem stating the conditions for the inverse of apartial function to be a hylomorphism is presented and proved. The theorem is appliedto many examples, including the classical problem of rebuilding a binary tree from itspreorder and inorder traversals.This thesis also investigates into the interplay between the above theorem and pre-vious results on optimisation problems. A greedy linear-time algorithm is derived forone of its instances — to build a tree of minimum height. The necessary monotonicitycondition, though looking intuitive, is difficult to establish. For general optimal bracket-ing problems, however, the thinning strategy gives an exponential-time algorithm. Thereason and possible improvements are discussed in a comparison with the traditional dy-namic programming approach. The greedy theorem is also generalised to a generic formallowing mutually defined algebras. The generalised theorem is applied to the optimalmarking problem defined on non-polynomial based datatypes. This approach deliverspolynomial-time algorithms without the need to convert the inputs to polynomial baseddatatypes, which is sometimes not convenient to do.The many techniques are applied to solve the Countdown problem, a problem de-rived from the popular television program of the same name. Different strategies towardderiving an efficient algorithm are experimented and compared.Finally, it is shown how to derive from its specification the inverse of the Burrows-Wheeler transform, a string-to-string transform useful in compression. Not only do weidentify the key property why the inverse algorithm works, but, as a bonus, we also outlinehow two generalisations of the transform may be derived.Most of the main part of this thesis has been published or is under preparation to bepublished. Chapter 4 and Chapter 7 have been published as [68] and [19] respectively.Journal versions of these articles have been submitted. The first two sections of Chapter5 has been published in [18]. The full proof in Section 5.1.2, however, is only seen in thisthesis. We are currently preparing an article covering Chapter 6.